Hilbert-Type Inequalities: Operators, Compositions and Extensions PDF Download

Author: Bicheng Yang
Publisher: Scientific Research Publishing, Inc. USA
ISBN: 1618969498
Category : Antiques & Collectibles
Languages : en
Pages : 410

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Book Description
Hilbert-type inequalities include Hilbert's inequalities, Hardy-Hilbert-type inequalities and Yang-Hilbert-type inequalities, which are important in Analysis and its applications.They are mainly divided three kinds of integral, discrete and half-discrete.In recent twenty years, there are many advances in research on Hilbert-type inequalities,especially in Yang-Hilbert-type inequalities. In this book, by using the way of weight functions, the parameterized idea and technique of Real and Functional Analysis, we introduce multi-parameters and provide three kinds of double Hilbert-type inequalities with the general measurable kernels and the best possible constant factors. The equivalent forms, the reverses and some particular inequalities are obtained. Furthermore, the operator expressions with the norm, a large number of examples on the norm, some composition formulas of the operators, and three kinds of compositional inequalities with the best possible constant factors are considered. The theory of double Hilbert-type inequalities and operators are almost built. The lemmas and theorems provide an extensive account of these kinds of inequalities and operators.

Author: CV-Bicheng Yang
Publisher: Scientific Research Publishing, Inc. USA
ISBN: 1649977778
Category : Antiques & Collectibles
Languages : en
Pages : 189

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Book Description
In this book, applying the weight functions, the idea of introduced parameters and the techniques of real analysis and functional analysis, we provide a new kind of half-discrete Hilbert-type inequalities named in Mulholland-type inequality. Then, we consider its several applications involving the derivative function of higher-order or the multiple upper limit function. Some new reverses with the partial sums are obtained. We also consider some half-discrete Hardy-Hilbert’s inequalities with two internal variables involving one derivative function or one upper limit function in the last chapter. The lemmas and theorems provide an extensive account of these kinds of half-discrete inequalities and operators.

Author: Themistocles M. Rassias
Publisher: Springer
ISBN: 1493912461
Category : Mathematics
Languages : en
Pages : 555

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Book Description
As Richard Bellman has so elegantly stated at the Second International Conference on General Inequalities (Oberwolfach, 1978), “There are three reasons for the study of inequalities: practical, theoretical, and aesthetic.” On the aesthetic aspects, he said, “As has been pointed out, beauty is in the eye of the beholder. However, it is generally agreed that certain pieces of music, art, or mathematics are beautiful. There is an elegance to inequalities that makes them very attractive.” The content of the Handbook focuses mainly on both old and recent developments on approximate homomorphisms, on a relation between the Hardy–Hilbert and the Gabriel inequality, generalized Hardy–Hilbert type inequalities on multiple weighted Orlicz spaces, half-discrete Hilbert-type inequalities, on affine mappings, on contractive operators, on multiplicative Ostrowski and trapezoid inequalities, Ostrowski type inequalities for the Riemann–Stieltjes integral, means and related functional inequalities, Weighted Gini means, controlled additive relations, Szasz–Mirakyan operators, extremal problems in polynomials and entire functions, applications of functional equations to Dirichlet problem for doubly connected domains, nonlinear elliptic problems depending on parameters, on strongly convex functions, as well as applications to some new algorithms for solving general equilibrium problems, inequalities for the Fisher’s information measures, financial networks, mathematical models of mechanical fields in media with inclusions and holes.

Author: Bicheng Yang
Publisher: Springer Nature
ISBN: 3030292681
Category : Mathematics
Languages : en
Pages : 152

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Book Description
This book is aimed toward graduate students and researchers in mathematics, physics and engineering interested in the latest developments in analytic inequalities, Hilbert-Type and Hardy-Type integral inequalities, and their applications. Theories, methods, and techniques of real analysis and functional analysis are applied to equivalent formulations of Hilbert-type inequalities, Hardy-type integral inequalities as well as their parameterized reverses. Special cases of these integral inequalities across an entire plane are considered and explained. Operator expressions with the norm and some particular analytic inequalities are detailed through several lemmas and theorems to provide an extensive account of inequalities and operators.

Author: Bicheng Yang
Publisher: Scientific Research Publishing, Inc. USA
ISBN: 1618968262
Category : Antiques & Collectibles
Languages : en
Pages : 273

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Book Description
In 1934, G. H. Hardy et al. published a famous book entitled “Inequalities”, in which a theory about Hardy-Hilbert-type inequalities with the general homogeneous kernels of degree-1 and the best possible constant factors was built by introducing one pair of conjugate exponents. In January 2009, for generalized theory of Hardy-Hilbert-type inequalities, a book entitled “The Norm of Operator and Hilbert-Type Inequalities” (by Bicheng Yang) was published by Science Press of China, which considered the theory of Hilbert-type inequalities and operators with the homogeneous kernels of degree negative numbers and the best possible constant factors, by introducing two pairs of conjugate exponents and a few independent parameters. In October 2009 and January 2011, two books entitled “Hilbert-Type Integral Inequalities” and “Discrete Hilbert-Type Inequalities” (by Bicheng Yang) were published by Bentham Science Publishers Ltd., which considered mainly Hilbert-type integral and discrete inequalities with the homogeneous kernels of degree real numbers and applications. In 2012, a book entitled “Nonlinear Analysis: Stability, Approximation, and Inequality” was published by Springer, which contained Chapter 42 entitled “Hilbert-Type Operator: Norms and Inequalities” (by Bicheng Yang). In this chapter, the author defined a general Yang-Hilbert-type integral operator and studied six particular kinds of this operator with different measurable kernels in several normed spaces. In 2014, a book entitled “Half-Discrete Hilbert-Type Inequalities” was published in World Scientific Publishing Co. Pte. Ltd. (in Singapore), in which, the authors Bicheng Yang and L. Debnath considered some kinds of half-discrete Yang-Hilbert-type inequalities and their applications. In a word, the theory of Hilbert-type integral, discrete and half- discrete inequalities is almost built by Bicheng Yang et al. in the above stated books.

Author: Gradimir V. Milovanović
Publisher: Springer
ISBN: 149390258X
Category : Mathematics
Languages : en
Pages : 873

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Book Description
This book, in honor of Hari M. Srivastava, discusses essential developments in mathematical research in a variety of problems. It contains thirty-five articles, written by eminent scientists from the international mathematical community, including both research and survey works. Subjects covered include analytic number theory, combinatorics, special sequences of numbers and polynomials, analytic inequalities and applications, approximation of functions and quadratures, orthogonality and special and complex functions. The mathematical results and open problems discussed in this book are presented in a simple and self-contained manner. The book contains an overview of old and new results, methods, and theories toward the solution of longstanding problems in a wide scientific field, as well as new results in rapidly progressing areas of research. The book will be useful for researchers and graduate students in the fields of mathematics, physics and other computational and applied sciences.

Author: Bicheng Yang
Publisher: World Scientific
ISBN: 9811267111
Category : Mathematics
Languages : en
Pages : 203

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Book Description
Hilbert-type inequalities, including Hilbert's inequalities proved in 1908, Hardy-Hilbert-type inequalities proved in 1934, and Yang-Hilbert-type inequalities first proved around 1998, play an important role in analysis and its applications. These inequalities are mainly divided in three classes: integral, discrete and half-discrete. During the last twenty years, there have been many research advances on Hilbert-type inequalities, and especially on Yang-Hilbert-type inequalities.In the present monograph, applying weight functions, the idea of parametrization as well as techniques of real analysis and functional analysis, we prove some new Hilbert-type integral inequalities as well as their reverses with parameters. These inequalities constitute extensions of the well-known Hardy-Hilbert integral inequality. The equivalent forms and some equivalent statements of the best possible constant factors associated with several parameters are considered. Furthermore, we also obtain the operator expressions with the norm and some particular inequalities involving the Riemann-zeta function and the Hurwitz-zeta function. In the form of applications, by means of the beta function and the gamma function, we use the extended Hardy-Hilbert integral inequalities to consider several Hilbert-type integral inequalities involving derivative functions and upper limit functions. In the last chapter, we consider the case of Hardy-type integral inequalities. The lemmas and theorems within provide an extensive account of these kinds of integral inequalities and operators.Efforts have been made for this monograph hopefully to be useful, especially to graduate students of mathematics, physics and engineering, as well as researchers in these domains.

Author: Bicheng Yang
Publisher: Scientific Research Publishing, Inc. USA
ISBN: 1649974094
Category : Antiques & Collectibles
Languages : en
Pages : 162

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Book Description
Hilbert-type inequalities including Hilbert’s inequalities (built-in 1908), Hardy-Hilbert-type inequalities (built-in 1934), and Yang-Hilbert-type inequalities (built-in 1998) played an important role in analysis and their applications, which are mainly divided into three classes of integral, discrete and half-discrete. In recent twenty years, there are many advances in research on Hilbert-type inequalities, especially in Yang-Hilbert-type inequalities. In this book, applying the weight functions, the parameterized idea, and the techniques of real analysis and functional analysis, we provide three kinds of Hilbert-type and Hardy-type integral inequalities in the whole plane as well as their reverses with parameters, which are extensions of Hilbert-type and Hardy-type integral inequalities in the first quarter. The equivalent forms, the operator expressions, and some equivalent statements of the best possible constant factors related to several parameters are considered. The lemmas and theorems provide an extensive account of these kinds of integral inequalities and operators. There are seven chapters in this book. In Chapter 1, we introduce some recent developments of Hilbert-type integral, discrete, and half-discrete inequalities. In Chapters 2-3, by using the weight function and real analysis, some new Hilbert-type and Hardy-type integral inequalities in the whole plane with the non-homogeneous kernel are given, and the cases of the homogeneous kernel are deduced. The equivalent forms and some equivalent statements of the best possible constant factors related to several parameters are obtained. We also consider the operator expressions as well as the reverses. In Chapters 4-7, the other two kinds of Hilbert-type and Hardy-type integral inequalities in the whole plane are also considered. We hope that this monograph will prove to be useful especially to graduate students of mathematics, physics, and engineering sciences.

Author: Panos M. Pardalos
Publisher: Springer
ISBN: 3319313177
Category : Mathematics
Languages : en
Pages : 754

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Book Description
The contributions in this volume aim to deepen understanding of some of the current research problems and theories in modern topics such as calculus of variations, optimization theory, complex analysis, real analysis, differential equations, and geometry. Applications to these areas of mathematics are presented within the broad spectrum of research in Engineering Science with particular emphasis on equilibrium problems, complexity in numerical optimization, dynamical systems, non-smooth optimization, complex network analysis, statistical models and data mining, and energy systems. Additional emphasis is given to interdisciplinary research, although subjects are treated in a unified and self-contained manner. The presentation of methods, theory and applications makes this tribute an invaluable reference for teachers, researchers, and other professionals interested in pure and applied research, philosophy of mathematics, and mathematics education. Some review papers published in this volume will be particularly useful for a broader audience of readers as well as for graduate students who search for the latest information. ​ Constantin Carathéodory’s wide-ranging influence in the international mathematical community was seen during the first Fields Medals awards at the International Congress of Mathematicians, Oslo, 1936. Two medals were awarded, one to Lars V. Ahlfors and one to Jesse Douglass. It was Carathéodory who presented both their works during the opening of the International Congress. This volume contains significant papers in Science and Engineering dedicated to the memory of Constantin Carathéodory and the spirit of his mathematical influence.

Author: A. Defant
Publisher: Elsevier
ISBN: 0080872875
Category : Mathematics
Languages : en
Pages : 579

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Book Description
The three chapters of this book are entitled Basic Concepts, Tensor Norms, and Special Topics. The first may serve as part of an introductory course in Functional Analysis since it shows the powerful use of the projective and injective tensor norms, as well as the basics of the theory of operator ideals. The second chapter is the main part of the book: it presents the theory of tensor norms as designed by Grothendieck in the Resumé and deals with the relation between tensor norms and operator ideals. The last chapter deals with special questions. Each section is accompanied by a series of exercises.